# Definition:Weierstrass Function

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## Contents

## Definition

A **Weierstrass function** is a real function defined on a closed real interval $I$ which is:

- $(1):\quad$ continuous on $I$
- $(2):\quad$ nowhere differentiable on $I$.

## Also see

## Historical Note

Karl Weierstrass first discussed a real function which was continuous everywhere but differentiable nowhere in his lectures in $1861$.

The function he initially demonstrated was defined as the sum of a Fourier Series:

- $\displaystyle f \left({x}\right) = \sum_{n \mathop \ge 0} a^n \cos \left({b^n \pi x}\right)$

where:

- $0 < a < 1$
- $b$ is a (strictly) positive odd integer

such that:

- $a b > 1 + \dfrac 3 2 \pi$

It did not appear in print until one of his students published it (with Weierstrass's permission) in $1874$.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($1815$ – $1897$)